Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville

2024/01/25 by Barinder S. Banwait, Maarten Derickx, Banwait, Barinder S. +1
Mathematics · Arts and Humanities · #Algebraic Geometry and Number Theory #Historical Studies and Socio-cultural Analysis #French Historical and Cultural Studies

paper · pdf · doi:10.48550/arxiv.2401.14514

Abstract

We study the problem of determining the groups that can arise as the torsion subgroup of an elliptic curve over a fixed quadratic field, building on work of Kamienny-Najman, Krumm, and Trbović. By employing techniques to study rational points on curves developed by Bruin and Stoll, we determine the possible torsion subgroups of elliptic curves over quadratic fields ℚ(√(d)) for all squarefree d with |d| < 800, improving on the previously known range of -5 < d < 26. We use our computations to study the validity of a conjecture of Granville concerning how many twists of a given hyperelliptic curve admit a nontrivial rational point.

Related

• Torsion groups of elliptic curves over quadratic fields ℚ(√(d)), 0
• Torsion of Elliptic Curves over Quadratic Fields
• Torsion of Rational Elliptic Curves over the ℤ_p-Extensions of Quadratic Fields
• Torsion of \mathbb Q-curves over number fields of small odd prime degree
• On Torsion Subgroups of Elliptic Curves over Quartic, Quintic and Sextic Number Fields